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Can anyone please help me to find the state space equation of this circuit diagram in the form of x=Ax+Bu?

The equation I am getting is not giving me square matrix. Can anyone help please?

enter image description here

These are the steps I have followed:

$$(I-Ic)Z+IZg+V=0$$

$$IZ-IcZ+IZg+V=0$$

$$Ldi/dt+RI+Lgdi/dt+RgI-Ldic/dt-RIc+V=0$$

$$(L+Lg)di/dt-Ldic/dt+(R+Rg)I-RIc+V=0$$

So you can see that I am getting two derivatives in the same equation, which makes them dependent on each other.

SamGibson
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safi2016
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    Please clearly label the currents and voltages (with symbols and sign convention marked clearly). Please show us the steps that you have followed (equations included) so that we may know where your calculation might have gone wrong. Also, how many *independent* state variables do you think this circuits has and why ? – AJN Oct 01 '21 at 11:34
  • Is 'A' matrix of order 1×1? – user215805 Oct 03 '21 at 19:34
  • Yes with two state variables – safi2016 Oct 04 '21 at 01:53
  • I'm not sure but seems to me that there is only one state variable i.e either I or Ic-I, and we can write our solution in form of d/dt(X) = AX + BU + B' d/dt (U) so the answer should be d/dt(I) =Rg+R/(Lg+L) I + [-1/(Lg+L) R/(Lg+L) ] U + [0 L] d/dt U where U is column vector of inputs V and Ic, can you check from answer sheet whether this is right or wrong? – user215805 Oct 04 '21 at 08:32
  • thanks for your suggestion. I am trying develop state space model for grid ties inverter. I haven't seen any work on that so I cann't check whether this is right or wrong. Again, I did not see this format d/dt(X) = AX + BU + B' d/dt (U) before. can you please refer any article or book where I can find this. – safi2016 Oct 04 '21 at 23:50
  • how can I manipulate the above equations to get state space model for Ic-I? Thanks – safi2016 Oct 05 '21 at 00:19
  • https://www.hindawi.com/journals/mpe/2019/1394725/ – user215805 Oct 05 '21 at 11:48
  • Check example 3 of above journal, I don't get notifications unless you tag me so you can use **@user215805** in comment so that I'll get notification of your commet – user215805 Oct 05 '21 at 11:51
  • @user215805 Thanks. I think I will go with I as state variable at the moment. what is the significance of derivative of an input in terms of stability in control system – safi2016 Oct 06 '21 at 23:36

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